Theorems · Theorem · commutative algebra
IsLocalRing.finite_quotient_iff
∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsLocalRing R] [IsNoetherianRing R] [Finite (IsLocalRing.ResidueField R)]
{I : Ideal R}, Finite (R ⧸ I) ↔ ∃ n, IsLocalRing.maximalIdeal R ^ n ≤ I- Defined in
- Mathlib.RingTheory.LocalRing.Quotient
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 102 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites15
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- CommRingstatement and proof · cited by 17,173
- Idealstatement and proof · cited by 4,748
- Finitestatement and proof · cited by 3,029
- HasQuotient.Quotientstatement and proof · cited by 2,301
- IsLocalRingstatement and proof · cited by 339
- IsLocalRing.maximalIdealstatement and proof · cited by 297
- IsNoetherianRingstatement and proof · cited by 268
- IsLocalRing.ResidueFieldstatement and proof · cited by 156
- Ideal.Quotient.factorproof · cited by 33
- IsNoetherian.noetherianproof · cited by 32
- Finite.of_surjectiveproof · cited by 14
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